I'm still worried about the cross-correlation of band-limited noise and my
colleague suggested a reductio ad absurdum argument to show that there
must be some correlation at different delays:
Compare the cross-correlation at two delays Rxy(t1) and Rxy(t2), where X
and Y are band-limited noise. Rxy(t) is zero -that's true when integrated
for an infinite length of time. When it's integrated over a finite time,
Rxy(t) is a zero-mean random number.
But what are the correlation between the cross-correlations Rxy(t1) and
Rxy(t2)?
When t1 and t2 are different, Rxy(t1) and Rxy(t2) are completely
independent random numbers -as you'd expect. If you now move t2 towards
t1, in the infinitesimal limit, Rxy(t1) and Rxy(t2 = t1 + dt) must
increasingly become dependent.
The bandlimited noise X and Y are generated by convolving white Gaussian
noise sequences with the filter function h(t) -a sinc function with a cos
carrier wave. In a hand-wavy kind of way, the random variable Rxy(t1) is
correlated with Rxy(t2) by the tail of h(t). When t2 = t1, the correlation
is 1 at the central peak of the sinc function.
With White Gaussian Noise (WGN), there'll be no correlation between the
finite cross-correlation functions unless t1 = t2 because the filter
function of h(t) of WGN is a delta function.
***
In summary, the measured cross-correlation over finite time at different
delays are not independent for band-limited noise.
***
Any comments?
tak
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Reply by tk229●April 2, 20052005-04-02

>the cross-correlation of two independent random processes is the product

of

>each mean. i'm pretty sure of that.

Very good point!
The cross-correlation func, Rxy(T) = E[ X(t)Y(t+T) ] can be written
separately if X and Y are statistically independent and wide-sense
stationary
Rxy(T) = E[X(t)] E[Y(t+T)] = MEAN[X] MEAN[Y], as you say.
Since X and Y have zero means, the cross-correlation Rxy(T) = 0. This is
also the condition for X and Y to be orthogonal.
Thanks for clearing that up!
Now the question remains about the cross-correlation function of
band-limited noise... I've made a little bit of pregress with this
problem. Here's how far I've got:
According to p. 176 of Peebles (3rd ed), it's possible to set an upper
limit on the cross correlation function:
ABS( Rxy(T) ) <= SQRT[ Rxx(0) Ryy(0) ]
If X and Y have the same bandwidth and statistical properties, Rxx(0) =
Ryy(0). If X and Y are band-limited noise, the autocorrelation Rxx(T) is a
cos function with a sinc envelope. Rxx(T) peaks at T=0 so I guess at least
this sets an upper limit.
One of the other question I had was about how you can talk about noise
being even or odd. Well, I think it was talking about the odd or evenness
of the power density spectrum (PDS). The PDS is given by,
p_xx(w) = Lim(T->inf) E[ ABS( X(w) )^2 ] / 2T
This means that p_xx(w) must be real and even. This seems to suggest that
the cross-correlation of band-limited noise Rxy(T), must be zero from the
arguments summarised in the original posting. From the Wiener-Khinchin
relation, the cross-power density spectrum p_xy(w) = 0.
Does this conclusion seem reasonable??
tak
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Reply by robert bristow-johnson●March 31, 20052005-03-31

in article 4d2dnXsG5PTz1tHfRVn-pQ@giganews.com, tk229 at tk229@cam.ac.uk
wrote on 03/31/2005 14:11:

> Correction:
>
>> Peyton & Peebles ("Probability, Random Variables and Random Signal
>> Principles", 3rd ed, p257):
>
> .....Should be Peyton Z Peebles -one person.
>
>
> On further thoughts, am I right in thinking that "a noise N(t) with an
> even spectrum" is equivalent to the cosine parts of the narrowband noise
>
> N(t) = A(t) [cos( Q(t) ) cos(w0 t) - sin( Q(t) ) sin(w0 t)]
>
> where the amplitude A(t) has a Reyleight prob distribution, the phase Q(t)
> has uniform prob distribution and w0 is the narrowband frequency?
>
>
> I still find it hard to believe how the cross correlation of two
> independent noise processes could have non-zero cross-correlation....
>
>
> Any comments?

the cross-correlation of two independent random processes is the product of
each mean. i'm pretty sure of that. sometimes a random process is input to
a linear system and the output of that is cross-correlated to the input and
that is not generally zero.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge."

Reply by tk229●March 31, 20052005-03-31

Correction:

>Peyton & Peebles ("Probability, Random Variables and Random Signal
>Principles", 3rd ed, p257):

.....Should be Peyton Z Peebles -one person.
On further thoughts, am I right in thinking that "a noise N(t) with an
even spectrum" is equivalent to the cosine parts of the narrowband noise
N(t) = A(t) [cos( Q(t) ) cos(w0 t) - sin( Q(t) ) sin(w0 t)]
where the amplitude A(t) has a Reyleight prob distribution, the phase Q(t)
has uniform prob distribution and w0 is the narrowband frequency?
I still find it hard to believe how the cross correlation of two
independent noise processes could have non-zero cross-correlation....
Any comments?
tak
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Reply by tk229●March 30, 20052005-03-30

Hi
I'm looking at an instrument used in radio astronomy and hoped that you
guys might be able to clarify something for me...
1) What is the cross-correlation function of two sets independent sets of
band-limited white noise?
2) Peyton & Peebles says that if the two sets of band-limited noise are
"even", Rxy(t) = 0 for all t because X and Y are orthogonal. How can you
start talking about band-limited noise being even or odd? What's it going
to be in a real RF system?
--------------------------------------
For reference, our system is 6-12GHz IF. I found an interesting plot in
Peyton & Peebles ("Probability, Random Variables and Random Signal
Principles", 3rd ed, p257):
It derives the cross correlation function of two band-limited function
centred about
w0 - W1 to w0 + W2
and -w0 -W1 to -w0 -W2
Basically the bandwidth is W = W1 + W2. If the two random processes are
even, W1 = W2 = W/2 (ie bands centred about +/-w0) and the two processes
are orthogonal. In this case, the cross-correlation function is naturally
zero. But are band-limited noise even? How can you talk about noise being
even or odd? W1 and W2 define the odd-ness of the noise.
They go on to show that the cross-correlation function is:
Rxy(t) = WP/pi sinc(Wt/2) sin[(W-2W1)t / 2],
where t is the lag. If W1 = W/2 (even case), Rxy(t) = 0. But this result
is kind of interesting because the correlated signal that we get is:
Rss(t) = W/pi sinc(Wt) cos(Wt)
------------------------------------------
Many thanks in advance
tak
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